Chapter 3 The theories used

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Chapter 3 The theories used

The aim of this study is the analysis of the human walk with and without the prosthesis in order to understand how to improve the prosthesis and the mathematical model. So it’s necessary to know and study the dynamic of the walker and in particular the act of walking in terms of mathematical study and the dynamic of the prosthesis. The field of the Engineering Dynamics used in this research is the Multi-body dynamics.

The Multi-body dynamics is a specific field of engineering dynamics dealing with the mathematical modelling, the simulation and the analysis of structural system made of subcomponents (rigid or flexible bodies) interconnected through links called joints. Two main hypothesis have been made:

• to consider all the element(body) as rigid, so there is no strain for big

displacements and rotations;

• to consider all the links not perfect, so they will dissipate energy.

In this chapter it will be explained synthetically the theory used, the Lagrange’s equations, the numerical scheme used for the resolution and at last the definition of the unilateral constraints [8].

3.1. Lagrange’s equations

Two different approaches to apply the Newton’s laws could have been chosen:

• The Principle of the virtual work.

• The Lagrange’s equations.

If on one side the Principle of the virtual work gives a good physical interpretation in the sense of Newton’s laws on the other it’s more difficult to model itself for the program. Moreover, the forces on the different parties of the body can also be calculated by the Lagrange coefficients. Indeed, the principal idea of the resolution is to calculate in symbolic the energy of each part and then to derive the equations to obtain the right matrix.

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3.1.1. Definition of the Lagrange’s equations

To obtain the Lagrange’s equations, we need to have the kinetic energy. For a N parts system, the kinetic energy is:

N N k k k k ik k k i T m u u m u = = = =

∑

⋅ =

∑∑

3 ⋅ 2 1 1 1 1 1 2 2 (3.1)

As a consequence, the inertia forces associated to the displacement uik can be

derived from k ik ik d T m u dt u ⎛ ∂ ⎞₌ ⎜_∂ ⎟ ⎝ ⎠ (3.2)

Writing the Principle of the virtual work for N particles ,you obtain

(

)

N k ik ik ik k i m u f u = = − ⋅ δ =

∑ ∑

3 1 1 0 (3.3) where fik are the components of a force field f , in an inertial frame; m is a mass ; u ik

represents its acceleration and δ is the virtual displacement of the particle. u_ik But it is known that

n ik ik s s s u u q q =1 ∂ δ = δ ∂

∑

(3.4)

where δ is the virtual displacement written in function of the generalized u_ik

coordinates q( q₁………..q ). The coefficients s

ik s

u q ∂

∂ define the displacement

directions of mass k when the generalized coordinate q is varied.

By substituting the equations (3.2) and (3.4) in the equation (3.3), you obtain

n N ik ik s s k i ik s u d T f q dt u q = = = ⎛ ⎛ ∂ ⎞₋ ⎞_⋅∂ _{δ =} ⎜ ⎜ ⎟ ⎟ ⎜ _⎝_∂ _⎠ ⎟ _∂ ⎝ ⎠

∑∑∑

3 1 1 1 0 (3.5) By developing and using the following equation:

ik ik ik ik s ik s ik s u u u d T d T T d dt u q dt u q u dt q ⎛ ∂ _⋅∂ ⎞₌ ⎛ ∂ ⎞_⋅∂ ₊ ∂ _⋅ ⎛∂ ⎞ ⎜_∂ _∂ ⎟ ⎜_∂ ⎟ _∂ _∂ ⎜_∂ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (3.6)

The equation (3.5) becomes:

n N ik ik ik ik s u u u d T T d f q ⎛ ⎛ ∂ ∂ ⎞ ∂ ⎛∂ ⎞ ∂ ⎞ ⋅ − − δ = ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ _∂ _∂ _∂ _∂ _∂ ⎟

∑∑∑

3 0 (3.7)

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Now you want to find a way to simplify the equation (3.7), and, for that reason, find a relation between ( ik s u q ∂ ∂ ) and ( ik s u q ∂

∂ ).The differential of u is: ik n ik ik ik s s s u u u q t = q ∂ ∂ = + ∂

∑

1 ∂ (3.8) This implies: ik ik s s u u q q ∂ ∂ = ∂ ∂ (3.9)

Substituting this result into the first term of (3.7) and recalling the chain rule for

derivation, the equation of motion associated to the degree of freedom q now takes _s

the form n s s s s s d T T Q q dt q q 1 0 = ⎛ ∂ ₋ ∂ ₋ ⎞_{⋅ δ =} ⎜ _∂ _∂ ⎟ ⎝ ⎠

∑

(3.10)

where Qs represents the generalized force associated to q and it is given by s N ik s ik k i s u Q f q 3 1 1 = = ∂ = ∂

∑∑

(3.11) Since (3.10) must be satisfied for any δ , you find that q_s

s s s d T T Q dt q q 0 ∂ ∂ − − = ∂ ∂ s=1,…….n (3.12)

The equations (3.12) define the Lagrange equations.

In the case where a potential V can be defined such that the forces are derived as

ik ik V f u ∂ = − ∂ (3.13) the generalized forces can be obtained as

N ik s k i ik s s u V V Q u q q 3 1 1 = = ∂ ∂ ∂ = − = − ∂ ∂ ∂

∑∑

(3.14) In that case it can be shown that the total energy is conserved and the forces are said conservative.

If the system is conservative, that is to say that we can write the forces from the potential, the equation of the movement or equation of Lagrange is:

s s s d T T V dt q q q 0 ∂ ∂ ∂ − + = ∂ ∂ ∂ s=1,………n (3.15)

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In the case where conservative and not-conservative forces are applied on a system, the Lagrange equations state that the dynamic equilibrium of the system is expressed in terms of its generalized coordinates by the n equations

s s s s d T T V Q dt q q q 0 ∂ ₋ ∂ ₊ ∂ ₋ ₌ ∂ ∂ ∂ s=1,………n (3.16)

3.1.2. Dissipation in Lagrange’s equation

The Lagrange’s equations are based on a energetic approach. So when there is a lost of energy due to dissipation forces (a damper for example), the Lagrange’s equation are modified.

A dissipation force depends on the speed of the particle. The dissipation force can thus expressed by:

ik ik k k k k v X C f v v ( ) = − (3.17) where • Ck is a constant

• fk (vk) is the dissipation function depending on the velocity • vk is thevelocity of the particle

• vik is the coordinate of the velocity vector

To consider the dissipation forces in the Lagrange’s equation (3.16), the Principle of virtual works is used again for this type of force:

n N N n ik s s ik ik ik s s i k i k s s u Q q X u X q q 3 3 1 1 1 1 1 1 = = = = = = ∂ δτ = ⋅ δ = ⋅ δ = ⋅ δ ∂

∑

∑∑

∑∑∑

(3.18)

By using the expression (3.17) in the previous equation, you obtain

( )

N ik ik s k k k i k k s v u Q C f v v q 3 1 1 = = ∂ = − ∂

∑∑

(3.19) But by definition, you know that: u_ik =v_ik.You have thus by using the equation (3.9):

ik ik s s v u q q ∂ ∂ = ∂ ∂ (3.20)

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( )

N N N ik ik k k k s k k k k ik k k k i k k s k k s i k s v v f v v Q C f v C v C f v v q v q q 3 3 2 1 1 1 1 1 ( ) 1 ( ) 2 = = = = = ∂ ∂ ⎡ ⎤ ∂ = − = − _⎢ _⎥= − ∂ ∂ _⎣ _⎦ ∂

∑∑

∑

(3.21)

If you define the dissipation force as shown below

k v N k k k D C f d 1 0 ( ) = =

∑ ∫

γ γ (3.22) the term of the not-conservative forces contribute is given by

s s D Q q ∂ = − ∂ (3.23) In conclusion, the Lagrange’s equations of motion are now written with conservative, not-conservative, and dissipation forces:

s s s s s d T T V D Q dt q q q q 0 ∂ ₋ ∂ ₊ ∂ ₊ ∂ ₋ ₌ ∂ ∂ ∂ ∂ (3.24)

3.1.3. Lagrange’s equation with constraints

One of the advantages of using the Lagrange’s equation is that the forces don’t explicitly appear, and thus ease the resolution and input data. However when some constraints are applied in the system it’s necessary to make the reaction forces appear explicitly in the expression of the equilibrium. First you have to define the typology of the constraint, and then present a way of the making them explicit in the expression of the Lagrange’s equation.

Kinematic constraint

The kinematic constraints reduce the motion possibilities of a system. In fact without kinematic constraints, the state of the system would be completely defined by the 3N displacement components u since, starting from a reference configuration _ik x , they _ik represent the instantaneous configuration

( )

(

)

ik t xik uik xjk,t

ξ = + i, j = 1,2,3 and k = 1, ,,,,,,,,,N (3.25) The system is then said to posses 3N degrees of freedom.

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• Holonomic constraints: they are defined as constraints that can be expressed as implicit relationships of the type:

(

ik

)

h ξ ,t =0 (3.26) This equation define a direct relationship between possible displacements and in particular you can say that every time a relation like (3.26) is expressed, the system loses one degree of freedom.

• Non holonomic constraint: they are constraints that can not be put in the

same form of the holonomic constraints. They are most often expressed as a differential relation of the form

(

)

nh

ik ik

h ξ ξ, ,t = (3.27) 0 Unlike holonomic constraints, these relationships can not be integrated and therefore do not allow to write an explicit relation between virtual displacements. In this case the constraints cannot decrease the number of the degrees of freedom of the system.

Lagrange’s multipliers

It’s necessary in this case to make the reaction forces appear explicitly in the expression of the equilibrium choosing generalized coordinates that not satisfy the kinematic constraints. You consider now a holonomic constraint and lets take its variation: N ik k i ik h h u u 3 1 1 0 = = ∂ δ = δ = ∂

∑∑

(3.28) The derivates ik h u ∂

∂ define the constrained directions to which the displacements

should be orthogonal in order to satisfy the constraint. Thus this reaction forces can be expressed as ik ik h R u ∂ = λ ∂ (3.29)

where λ describes the unknown intensity of the reaction and is called Lagrange

multiplier.

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N N ik ik ik k i k i ik h R u u h u 3 3 1 1 1 1 = = = = ∂ δ = λ δ = λδ ∂

∑∑

(3.30)

By choosing generalized coordinates q that do not satisfy the constraints h=0, that _s

is to say that the forces R could not be equal to zero, the equation of motion ik

deducted from the virtual work principle says that

N ik k ik ik k i ik s u h m u f u q 3 1 1 0 = = ⎛ ₋ _{− λ} ∂ ⎞ ∂ ₌ ⎜ _∂ ⎟ _∂ ⎝ ⎠

∑∑

(3.31)

and the reaction force intensity is determined to satisfy the condition

(

ik

)

(

ik ik

(

s

)

h ξ ,t =h x +u q t t, , = (3.32) 0 The Lagrange’s equations will then change considering the constraints too.

The Lagrange’s equations and the complementary kinematic condition are

(

)

(

)

s s s s s s ik ik s d T T V D h Q dt q q q q q h x u q t t 0 , , 0 ∂ ∂ ∂ ∂ ∂ ⎧ ₋ ₊ ₊ ₋ _{− λ} ₌ ⎪ ∂ ∂ ∂ ∂ ∂ ⎪⎪ ⎨ ⎪ ₊ ₌ ⎪ ⎪⎩ s=1,………..,n (3.33)

Just one constraints is here considered but more can be integrated by the same way.

3.2. Numerical resolution and time integration

3.2.1. Generalities about numerical schemes

The previous section was about the formation of the motion equations for the considered system. You will now discuss about how to solve these differential equations.

Principal equations

For a linear mechanical system, the fundamental equation of motion can generally be presented like:

( )

Mq+Cq+Kq =p t (3.34) The principle idea to solve this equation is to discrete the time and to solve the equation from one step time to the next one.

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The system, at the first step, is supposed to be known: the initial position

(

)

q₀ =q t =0 and initial speed q₀ =q t

(

=0

)

are given. The initial acceleration can easily be calculated:

( )

(

)

q₀ =M−1 p t₀ −Cq₀ −Kq₀ (3.35) At each step time, the dynamic equilibrium must be verified, that is to say:

( )

n n n n

Mq +Cq +Kq =p t (3.36) Obviously the q , _n q_n and q (the displacement, the velocity and the acceleration at _n the time t = ) are related by the following equations: t_n

(

)

(

)

n n n _t n n n t q q t t q t q q t t q t 0 0 lim lim ∆ → ∆ → ⎧ − ∆ = ⎪⎪ ±∆ ⎨ − ∆ ⎪ ₌ ⎪ _±∆ ⎩ ∓ ∓ (3.37) The time integration schema consists in approximating these limits by finite

differences in order to provide the system with two additional relations between q , _n

n

q and q . The system (3.37) is equivalent to a set of algebraic equations, that can _n

be solved by traditional solver.

Integration scheme

The system (3.34) is a system of second order, but using the so-called state-space vector can be reduced to a first-order. You therefore define and use the state-space vector of dimension 2n to reduce the system to a first-order.

( )

q t

( )

_{( )}

u t q t ⎛ ⎞ ⎜ ⎟ =_⎜ _⎟ ⎝ ⎠ (3.38)

By adding the trivial equation: Iq−Iq= to the system (3.34), it can be written by 0

following way:

( )

q q p t M C K I q I q 0 0 0 ₀ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⋅⎜ ⎟+ ⋅⎜ ⎟ ⎜= ⎟ ⎜ ⎟ _{⎜ ⎟} ⎜₋ ⎟ _{⎜ ⎟} _⎜ _⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (3.39) Or expressed in the next way by using the state vector u :

( )

M p t−1

⎛ ⎞

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where : M C M K A I 1 1 0 − − ⎛− − ⎞ = ⎜ ⎟ ⎝ ⎠ (3.41)

In this way the system is now put in the first order form.

Criteria of the quality of a numerical scheme

Several types of schemas have been developed, and can be controlled by the following criteria.

• Consistency

The consistency is a necessary condition for the convergence. To check that a schema is consistent means to verify that when the time step decreases, the numerical respond tends to an analytical solution. That can be summarized by the following equation:

( )

h u u t t u u t t 0 lim → ∆ ⎛ ⎞ ⎜ _∆ ⎟ ⎛ ⎞ = ⎜ ⎟ ⎜_⎜ ⎟_⎟ ∆ ⎜ _{⎟ ⎝} _⎠ ⎜ _∆ ⎟ ⎝ ⎠ (3.42)

The smaller the step time becomes, the closer the respond is from the real solution. The perfect solution would be thus obtained by a really small time step, however the calculus time depends on the time step.

So you have to find a compromise between the precision and the processing time.

• Stability

To explain it properly, you have to consider the following equation:

n n

u ₊₁ =A u* (3.43)

The matrix A* is an amplification matrix. The schema will be considered as

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the solution will be inexact and much bigger than what it should be. This problem may appear for physical system without damping or energy lost; on the contrary a physical damping may compensate the overvaluation of the solution and make the schema stable.

• Accuracy

Even though numerical schemas are converging, their accuracy is not guaranteed. A good way to check if the schema is stable or not is to look at the global energy. For conservative system, it must be constant. Bad accuracy may be perturb the global energy.

3.2.2. Numerical scheme used

The NewMark schema

In our case, the mechanical equilibrium equation (3.34) can be written as shown below :

(

) ( )

( )

T Mq f q q t B q h q , , 0 0 ⎧ ₊ ₊ _{λ =} ⎪ ⎨ = ⎪⎩ (3.44)

where the matrix M f B, , come from the Lagrange’s equation and they are equal to:

ss s s s s s s s s T M q h B q T T V D f q Q q q q q q 2 2 ∂ = ∂ ∂ = ∂ ⎛ ⎞ ∂ ∂ ∂ ∂ ∂ = _⎜ _⎟⋅ − + + − ∂ _⎝∂ _⎠ ∂ ∂ ∂ (3.45)

The main issue is however that the dynamic equations are not linear, due to the fact that f depends at the same time on q and q . In a first step, it’s necessary to proceed to the time discretization: the residual vector r , which represents the results of dynamic equations and of cinematic constraints, will be defined. This vector has to

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( )

(

) ( )

( ) ( )

T eq n n n n n n n cons n n n r q Mq f q q B q r q h q , 1 1 1 1 1 1 1 , 1 1 1 , + + + + + + + + + + ⎧ ₌ ₊ ₊ _λ ⎪ ⎨ = ⎪⎩ (3.46)

To make it converge to zero, the implicit-Newmark uses the Newton-Raphson convergence method. In this case you will thus use another iteration process to ensure the convergence. Each iteration will be noted by the following way:

( )

(

) ( )

( ) ( )

T k k k k k k eq n n n n n n n k k cons n n n r q Mq f q q B q r q h q , 1 1 1 1 1 1 1 , 1 1 1 , + + + + + + + + + + ⎧ ₌ ₊ ₊ _λ ⎪ ⎨ = ⎪⎩ (3.47)

a correction is provided at each iteration on the acceleration ∆ and on the qk

Lagrange’s coefficient ∆λ . The correction is made by the following way: k

k k k n n k k k n n k k k n n k k k n n q q q q q t q q q t q 1 1 1 1 2 1 1 1 2 1 1 1 1 1 + + + + + + + + + + + + ⎧ = + ∆ ⎪ = + β∆ ∆ ⎪⎪ ⎨ = + γ∆ ∆ ⎪ ⎪ λ = λ + ∆λ ⎪⎩ (3.48)

The residual vector evolves at each iteration:

( )

T k k k k k k eq n eq n n n k k k k cont n cont n n r r S q B q r r B q q 1 , 1 , 1 1 1 , 1 , 1 1 + + + + + + + + ⎧ ₊ _∆ ₊ _∆λ ⎪ ⎨ + ∆ ⎪⎩ (3.49)

where the iteration matrix k

n

S₊₁ is given by:

k t t

n

S₊₁=M+ γ∆tC + β∆t K2 (3.50) The matrix C and t K are defined by the following expressions: t

( )

(

)

t T t f C q B q f K q q ∂ ⎧ ₌ ⎪ _∂ ⎪ ⎨ _∂ _λ ∂ ⎪ ₌ ₊ ⎪ _∂ _∂ ⎩ (3.51)

The Lagrange’s equations are now a linear system at each time iteration:

( )

T k k k k n n eq n k k k cont n n S B q q r r t B q 1 1 , 1 2 , 1 1 0 / + + + + + ⎛ ⎞ _⎛_∆ _⎞ _⎛₋ _⎞ ⎜ ⎟_⋅_⎜ _⎟₌_⎜ _⎟ ⎜ ⎟ ⎜ ⎟ ⎜₋ _β∆ ⎟ ⎜ ⎟ ∆λ_⎝ _{⎠ ⎝} _⎠ ⎝ ⎠ (3.52)

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At the end of each spatial iteration, the correction will be fulfilled as presented in the equations (3.48). You will define a norm and a value under which you can consider the residual vector to be null, and thus break the spatial and dynamic iteration procedure. The time iteration will then take place.

This numerical schema is unconditionally stable for no-constraint system.

In this study you have to use a Newmark schema slightly modified: the Hilbert Hugues Taylor Methods or H.H.T. Methods.

Hilbert-Hugues-Taylor Method

The H.H.T, will use the same equations and resolution method, but will add some numerical damping. The equations will be solved for each time step, but the equations will consider not only current time iteration but also partly the equations of the previous time iteration through a coefficient α:

(

)

(

) ( )

)

(

) ( )

)

(

)

( )

T T n n n n n n n n n n n Mq f q q B q f q q B q h q h q 1 1 1 1 1 1 1 , , 0 1 0 + + + + + + ⎧ _{+ − α} ₊ _λ _{+ α} ₊ _{λ =} ⎪ ⎨ ⎪ − α + α = ⎩ (3.53)

The consistency is not modified by the α. Moreover, if the following relations are

verified, the schema becomes unconditionally stable:

(

)

2 1 0, 3 1 2 1 1 4 ⎧_{α ∈}⎡ ⎤ ⎪ ⎢_⎣ ⎥_⎦ ⎪ ⎪ γ = + α ⎨ ⎪ ⎪β = + α ⎪ ⎩ (3.54)

the two last ones are the Newmark conditions. α Will be taken at 5%.

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Initial parameters known: q₀ and q₀ Calculation of q₀ :

(

)

q₀ = −M f q q−1 ₀, ₀ Temporal iteration: t_n₌₁=t_n + ∆t Predictor:

(

)

(

)

n n n n n n n n n n n q q tq q q tq t q q q 1 2 1 1 1 1 0.5 + + + + = + − γ ∆ = + ∆ + −β ∆ = λ = λ

Calculus of the residual vector:

(

)

(

) (

)

(

)

( )

T T eq n n n n n n n n cont n n n r Mq f B f B r h q h q , 1 1 1 1 1 , 1 1 1 0 1 + + + + + + + = + − α + λ + α + λ = = − α + α Loop of Newton-Raphson YES NO

Calculation of the correction:

t t n n n S ₊₁ =M+ γ∆tC ₊₁+ β∆t K2 ₊₁

(

)

(

)

k k T eq n n n k k n cont n r q S B B r h , 1 1 1 2 1 _, ₁ 1 1 0 _/ + + + + ₊ ⎛− ⎞ ⎛∆ ⎞ ⎛ − α ⎞ ⎜ ⎟ ⋅⎜ ⎟= ⎜ ⎟ ⎜ − α ⎟ ⎜_∆λ ⎟ ⎜₋ _β ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ Fig.3.1: HHT method applied to a constrained non-linear system

Test on the convergence:

eq n r _, ₊₁ <εeq fn+1 cont n r _, ₊₁ <ε_cont Correction: n n n n n n n n q q q q q t q q q t q 1 1 2 1 1 2 1 1 1 1 + + + + + + + + = + ∆ = + β∆ ∆ = + γ∆ ∆ λ = λ + ∆λ

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3.3. The unilateral constraints

The numerical schema previously presented considers only constant constraints: the constraints are supposed to be known from the beginning, and will not change during the time iterations. However, in the walking process, some constraints may change, appear or disappear. They represent mostly the unilateral constraints and contacts. There are some methods to integrate them in the time iteration. In this research two main method will be presented: the use of the Lagrange’s multipliers and the Penalty method [9].Then in the next chapter the way of programming them in the model will be exposed.

3.3.1. Lagrange’s multipliers

Program of the constraints

The system to solve during the time iteration has been defined previously (3.44): there are two possible situations of numerical integration:

• the constraint is “on”

• the constraint is “off”

The constraint will be given at the beginning like a constant constraint. When the constraint is “off”, no force is applied to keep the constraint true: the force will thus be

imposed null, by controlling λconst to be equal to zero and by suppressing the

corresponding constraints from h and B . When the constraint will be activated, this control will be released, the force will evolve enabling the constraint to remain activated. This method is used in the program to consider the foot hitting the ground.

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The foot contact is perfect: no slip will be considered. The orientation of the axes during the walking is shown in the fig 3.2 and the origin is to be supposed on the ground. The constraint is programmed to fix the foot when it touches the ground. As long as the vertical position y of the foot is negative the foot is free of movement, and

const

λ will be maintained to zero. Once y_foot

( )

tn >0, the constraint is activated. At the next time step, you will have λ_const

(

tn+ ≠1

)

0.

Release of the constraint

The release of the constraint must also be programmed, because you want the model to make several steps. The sign of the force has to be observed, during the contact. If λconst< 0, the force prevents the foot from going into the ground. As long

as λ_const remains negative, the constraint has to be activated.

Once λ_const> 0, the force prevents the foot from taking off, the constraint has thus to be removed.

But, on the simplest model used, when the foot is modelled by a concentrated mass on the ankle, the constraint is not released in this way.

The hypothesis of the McGeer model are used: the two feet cannot touch the ground at the same time. So, as soon as a foot hits the ground, the constraint on the other is released.

Drawbacks of the this method

The problem about the utilization of the Lagrange’s multipliers is the definition of the time of the activation of the constraint. For this reason it’s necessary to take the smaller time step to minimize the error done.

The second problem is connected with the stability of the system: indeed with the activation of the constraint some elements of the Kt matrix could pass from the value of zero to an other different. This fact is showed by a big increase of the force in the system. If this increase is too big, there is the possibility of instability of the system. At last when several constraints are activated and deactivated, the size of the vector

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and will not be considered during the update of the activated constraints. However when lots of them have to be change, the update of the system might slow down the resolution of the equations, and also complicate the management of the multipliers of Lagrange.

3.3.2. The penalty method

Program of the constraint

This method is used in the program to avoid the hypertension of the leg: as the fig.3.3 shows when θ − θ > 0 there is a not natural movement of the leg and the leg ₂ ₁ is hypertensive. So it’s necessary to block this movement.

This method does not change the constraints,

Fig.3.3: The problem of the hypertension

As soon as the hypertension condition is detected and the constraint activated, the program will consider a new spring and a new damper applied on the degree of freedom corresponding as we can see on the fig. 3.4.

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So the stiffness and friction functions will be changed and the same for the function

f , which considers the force applied on the system, and therefore for the matrix K t

and t

C . The vector of constraint h and the matrix B will remain the same. Once the test stated that the constraint has been violated, the system is replaced right before the moment where there was the violation of the constraints and actives the constraint, by a linear interpolation, which is legitimate if the time step is taken small enough.

Release of the constraint

At each time step, a test will be made to check that the constraint has still reason to

be active. If the angle between the thigh and the shin is less than 180±, or if

2 1

θ − θ <0 the hyperextension is not a problem and the unilateral constraint can be suppressed. It will be suppressed at the current time step : then the program will recalculate the answer without constraint and then check out if it’s necessary to reactivate the constraint. In the next chapter the way of programming them will be exposed more in detail.

Drawbacks of this method

The spring will enable a slight violation of the constraint, due to the oscillations implied by the spring. This is also what can be observed in physical contact, that is there is a small penetration of the constraint.

But this methods has a big advantage because the value of the spring and damper is adjustable, and so it’s possible to change them to make the numerical response correspond to the physical one.